Equivalence relation

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2010 Mathematics Subject Classification: Primary: 03E [MSN][ZBL]

Let $X$ be a set. An equivalence relation on $X$ is a subset $R\subseteq X\times X$ that satisfies the following three properties:

1) Reflexivity: for all $x\in X$, $(x,x)\in R$;

2) Symmetry: for all $x,y\in X$, if $(x,y)\in R$ then $(y,x)\in R$;

3) Transitivity: for all $x,y,z \in X$, if $(x,y)\in R$ and $(y,z)\in R$ then $(x,z)\in R$.

When $(x,y)\in R$ we say that $x$ is equivalent to $y$.

Instead of $(x,y)\in R$, the notation $xRy$, or even $x\sim y$, is also used.

An equivalence relation is a binary relation.

Example: If $f$ maps the set $X$ into a set $Y$, then $R=\{(x_1,x_2)\in X\times X\,:\, f(x_1)=f(x_2)\}$ is an equivalence relation (cf. Kernel of a function).

For any $y\in X$ the subset of $X$ that consists of all $x$ that are equivalent to $y$ is called the equivalence class of $y$, denoted $[y]_R$. Any two equivalence classes either are disjoint or coincide, that is, any equivalence relation on $X$ defines a partition (decomposition) of $X$, and vice versa.

A subset $A$ of $X$ is saturated with respect to $R$ if $a \in A,\, b \in X,\,a R b \Rightarrow b \in A$: so $A$ is saturated if and only if it is a union of equivalence classes of $R$. For any subset $A$ of $X$, the saturation $A^R$ is the set $\{ b \in X : \exists a \in A,\,a R b \} = \cup_{a \in A} [a]_R$, the union of the equivalence classes that meet $A$.


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